2The Two-Good CaseThe types of relationships that can occur when there are only two goods are limitedBut this case can be illustrated with two-dimensional graphs

3Gross ComplementsQuantity of yx1x0y1y0U1U0When the price of y falls, the substitution effect may be so small that the consumer purchases more x and more yIn this case, we call x and y gross complementsx/py < 0Quantity of x

4Gross SubstitutesQuantity of yx1x0y1y0U0When the price of y falls, the substitution effect may be so large that the consumer purchases less x and more yU1In this case, we call x and y gross substitutesx/py > 0Quantity of x

5A Mathematical TreatmentThe change in x caused by changes in py can be shown by a Slutsky-type equationsubstitutioneffect (+)income effect(-) if x is normalcombined effect(ambiguous)

6Substitutes and ComplementsFor the case of many goods, we can generalize the Slutsky analysisfor any i or jthis implies that the change in the price of any good induces income and substitution effects that may change the quantity of every good demanded

7Substitutes and ComplementsTwo goods are substitutes if one good may replace the other in useexamples: tea & coffee, butter & margarineTwo goods are complements if they are used togetherexamples: coffee & cream, fish & chips

9Asymmetry of the Gross DefinitionsOne undesirable characteristic of the gross definitions of substitutes and complements is that they are not symmetricIt is possible for x1 to be a substitute for x2 and at the same time for x2 to be a complement of x1

12Asymmetry of the Gross DefinitionsInserting this into the budget constraint, we can find the Marshallian demand for ypyy = I – pyan increase in py causes a decline in spending on ysince px and I are unchanged, spending on x must rise ( x and y are gross substitutes)but spending on y is independent of px ( x and y are independent of one another)

14Net Substitutes and ComplementsThis definition looks only at the shape of the indifference curveThis definition is unambiguous because the definitions are perfectly symmetric

15Gross ComplementsEven though x and y are gross complements, they are net substitutesQuantity of ySince MRS is diminishing, the own-price substitution effect must be negative so the cross-price substitution effect must be positivey1y0U1U0x0x1Quantity of x

16Substitutability with Many GoodsOnce the utility-maximizing model is extended to may goods, a wide variety of demand patterns become possibleAccording to Hicks’ second law of demand, “most” goods must be substitutes

17Substitutability with Many GoodsTo prove this, we can start with the compensated demand functionxc(p1,…pn,V)Applying Euler’s theorem yields

18Substitutability with Many GoodsIn elasticity terms, we getSince the negativity of the substitution effect implies that eiic  0, it must be the case that

19Composite CommoditiesIn the most general case, an individual who consumes n goods will have demand functions that reflect n(n+1)/2 different substitution effectsIt is often convenient to group goods into larger aggregatesexamples: food, clothing, “all other goods”

20Composite Commodity TheoremSuppose that consumers choose among n goodsThe demand for x1 will depend on the prices of the other n-1 commoditiesIf all of these prices move together, it may make sense to lump them into a single composite commodity (y)

21Composite Commodity TheoremLet p20…pn0 represent the initial prices of these other commoditiesassume that they all vary together (so that the relative prices of x2…xn do not change)Define the composite commodity y to be total expenditures on x2…xn at the initial pricesy = p20x2 + p30x3 +…+ pn0xn

23Composite Commodity TheoremAs long as p20…pn0 move together, we can confine our examination of demand to choices between buying x1 and “everything else”The theorem makes no prediction about how choices of x2…xn behaveonly focuses on total spending on x2…xn

24Composite CommodityA composite commodity is a group of goods for which all prices move togetherThese goods can be treated as a single commoditythe individual behaves as if he is choosing between other goods and spending on this entire composite group

28Example: Composite CommodityIf we assume that the prices of housing services (py) and electricity (pz) move together, we can use their initial prices to define the “composite commodity” housing (h)h = 4y + 1zThe initial quantity of housing is the total spent on housing (75)

32Household Production ModelAssume that individuals do not receive utility directly from the goods they purchase in the marketUtility is received when the individual produces goods by combining market goods with time inputsraw beef and uncooked potatoes yield no utility until they are cooked together to produce stew

33Household Production ModelAssume that there are three goods that a person might want to purchase in the market: x, y, and zthese goods provide no direct utilitythese goods can be combined by the individual to produce either of two home-produced goods: a1 or a2the technology of this household production can be represented by a production function

34Household Production ModelThe individual’s goal is to choose x,y, and z so as to maximize utilityutility = U(a1,a2)subject to the production functionsa1 = f1(x,y,z)a2 = f2(x,y,z)and a financial budget constraintpxx + pyy + pzz = I

35Household Production ModelTwo important insights from this general model can be drawnbecause the production functions are measurable, households can be treated as “multi-product” firmsbecause consuming more a1 requires more use of x, y, and z, this activity has an opportunity cost in terms of the amount of a2 that can be produced

36The Linear Attributes ModelIn this model, it is the attributes of goods that provide utility to individualsEach good has a fixed set of attributesThe model assumes that the production equations for a1 and a2 have the forma1 = ax1x + ay1y + az1za2 = ax2x + ay2y + az2z

37The Linear Attributes ModelxThe ray 0x shows the combinations of a1 and a2available from successively larger amounts of good xa2yThe ray 0y shows the combinations ofa1 and a2 available from successivelylarger amounts of good yzThe ray 0z shows thecombinations of a1 anda2 available fromsuccessively largeramounts of good za1

38The Linear Attributes ModelIf the individual spends all of his or her income on good xx* = I/pxThat will yielda1* = ax1x* = (ax1I)/pxa2* = ax2x* = (ax2I)/px

39The Linear Attributes Modelx* is the combination of a1 and a2 that would beobtained if all income was spent on xx*a2xy*y* is the combination of a1 and a2 thatwould be obtained if all income wasspent on yyZ*z* is the combination ofa1 and a2 that would beobtained if all income wasspent on zza1

41The Linear Attributes ModelA utility-maximizing individual would neverconsume positive quantities of all threegoodsa2xIndividuals with a preference towarda1 will have indifference curves similarto U0 and will consume only y and zU0yIndividuals with a preferencetoward a0 will haveindifference curves similarto U1 and will consume onlyx and yU1za1

42The Linear Attributes ModelThe model predicts that corner solutions (where individuals consume zero amounts of some commodities) will be relatively commonespecially in cases where individuals attach value to fewer attributes than there are market goods to choose fromConsumption patterns may change abruptly if income, prices, or preferences change

43Important Points to Note:When there are only two goods, the income and substitution effects from the change in the price of one good (py) on the demand for another good (x) usually work in opposite directionsthe sign of x/py is ambiguousthe substitution effect is positivethe income effect is negative

44Important Points to Note:In cases of more than two goods, demand relationships can be specified in two waystwo goods are gross substitutes if xi /pj > 0 and gross complements if xi /pj < 0because these price effects include income effects, they may not be symmetricit is possible that xi /pj  xj /pi

45Important Points to Note:Focusing only on the substitution effects from price changes does provide a symmetric definitiontwo goods are net substitutes if xi c/pj > 0 and net complements if xi c/pj < 0because xic /pj = xjc /pi, there is no ambiguityHicks’ second law of demand shows that net substitutes are more prevalent

46Important Points to Note:If a group of goods has prices that always move in unison, expenditures on these goods can be treated as a “composite commodity” whose “price” is given by the size of the proportional change in the composite goods’ prices

47Important Points to Note:An alternative way to develop the theory of choice among market goods is to focus on the ways in which market goods are used in household production to yield utility-providing attributesthis may provide additional insights into relationships among goods